Lecture Notes: Difference of Squares

Introduction

• Practice session on difference of squares.
• Review of concepts from the previous lesson.

Basics of Difference of Squares

• Square Numbers:
  • A number raised to the power of 2.
  • Example: ( x^4 ) is a square number because ( x^2 \times x^2 = x^4 ).
  • 9 is a square number because ( 3 \times 3 = 9 ).
• Factoring Differences of Squares:
  • Example: ( x^4 - 9 = (x^2 + 3)(x^2 - 3) ).
  • General form: ( a^2 - b^2 = (a+b)(a-b) ).

Handling Exponents

• Multiplying with Exponents:
  • Add the exponents when multiplying similar bases.
  • Example: ( x^4 \times x^4 = x^8 ) because exponents are added.
• Factorization Example:
  • ( x^8 - 1 = (x^4 + 1)(x^4 - 1) ).

Further Factorization

• Breaking Down Further:
  • ( x^4 - 1 = (x^2 + 1)(x^2 - 1) ).
  • Continue factoring until no further difference of squares can be applied.

Common Errors and Tips

• Addition vs. Subtraction:
  • Only "difference of squares" (subtraction) can be factored this way.
  • Addition ((x^2 + a^2)) cannot be factored in the same manner.
• Common Factor First:
  • Always check for a common factor before proceeding with factorization.
  • Example: ( 2(x^2 - 16) ) can be further factored as ( 2(x+4)(x-4) ).

Examples

• Example 1:
  • ( x^8 - 1 = (x^4 + 1)(x^4 - 1) = (x^4 + 1)(x^2 + 1)(x^2 - 1) ).
• Example 2:
  • ( 7(x^2 - 4) = 7(x+2)(x-2) ).
• Example 3:
  • ( 3x(x^2 - 4) = 3x(x+2)(x-2) ).

Practice and Homework

• Apply the difference of squares method to new problems.
• Always remember to look for common factors.
• Understand when to stop factorization (e.g., when only addition is left).

Conclusion

• Consistent practice will help in mastering difference of squares.
• Always follow the steps: Look for common factors, check the sign, and then apply difference of squares.